I am agreeing with Marcia. There isn't any reason for soft brick bits to fly sideways, so check your lid for pieces falling off.You could put a shelf above your glazed ware, but chances are pretty good that you need to replace the lid.
TJR.

This is a long shot and probably not right. Is there anyway a vent for the kiln (If you have a vent) is pushing air instead of pulling it out. My Kiln is out in a garage so I don't need a vent so I am not sure how they are set up on an kiln. If is possible for it to be going backwards this could put dust into the air. Hope this isn't pure stupid. But it might be. oh well. Hope you find the solution. and Happy firing.

i would take a guess that there might have been some loose soft brick dust sitting on the top edge of the kiln walls (from lid rubbing it, or from user touching the brick when loading/unloading), then when you closed the lid the slight woosh of air blew some of the dust inward on the piece.

do you have pics of the firebrick contaminants? usually you can tell where the firebrick came from just from what it looks like - since many lids have a refractory mortar coating on them, the brick looks different than those found on the walls.

also, you said your kiln doesn't have a vent, but does it have vent intake holes in the lid?